A Complexity Measure for Active Learning in Multi-group Mean Estimation

We study a \emph{max-risk} objective for active learning in a multi-group mean estimation $d$-armed bandits: a learner adaptively allocates a budget of $T$ samples across $d$ groups to minimize the worst-case uncertainty index $\max_{k\in[d]}σ_k^2/n_k$, where $σ_k$ is the standard deviation of the distribution of arm $d$, and $n_k$ is the number of times arm $d$ is sampled. We develop a local minimax framework and prove the first general lower bound for this objective, valid for any finite-variance hypothesis class. The bound separates difficulty into three orthogonal factors: a \emph{budget} term, a \emph{heteroscedasticity} index measuring how unevenly the uncertainty is spread across arms, and a model-dependent complexity measure, the \emph{Variance Local Curvature} ($\mathrm{VLC}$), which captures how much information a local change of variance creates inside the hypothesis class. For smooth classes, the $\mathrm{VLC}$ is a reparametrization of a variance--Fisher information, with closed-form values for common families. Benchmarking against the strongest available upper bound shows near-optimality up to logarithmic factors in broad regimes, and pinpoints a systematic gap in highly heterogeneous instances. Our proof introduces two key ingredients: a loss-induced $\ell_1$ geometry on the decision space, and a representation-based instance generator that reduces hard-instance construction to an explicit random matrix calculation.

An active learning framework for multi-group mean estimation

We study a fundamental learning problem over multiple groups with unknown data distributions, where an analyst would like to learn the mean of each group. Moreover, we want to ensure that this data is collected in a relatively fair manner such that the noise of the estimate of each group is reasonable. In particular, we focus on settings where data are collected dynamically, which is important in adaptive experimentation for online platforms or adaptive clinical trials for healthcare. In our model, we employ an active learning framework to sequentially collect samples with bandit feedback, observing a sample in each period from the chosen group. After observing a sample, the analyst updates their estimate of the mean and variance of that group and chooses the next group accordingly. The analyst's objective is to dynamically collect samples to minimize the collective noise of the estimators, measured by the norm of the vector of variances of the mean estimators. We propose an algorithm, Variance-UCB, that sequentially selects groups according to an upper confidence bound on the variance estimate. We provide a general theoretical framework for providing efficient bounds on learning from any underlying distribution where the variances can be estimated reasonably. This framework yields upper bounds on regret that improve significantly upon all existing bounds, as well as a collection of new results for different objectives and distributions than those previously studied.

MNL-Bandit with Knapsacks

We consider a dynamic assortment selection problem where a seller has a fixed inventory of $N$ substitutable products and faces an unknown demand that arrives sequentially over $T$ periods. In each period, the seller needs to decide on the assortment of products (of cardinality at most $K$) to offer to the customers. The customer's response follows an unknown multinomial logit model (MNL) with parameters $v$. The goal of the seller is to maximize the total expected revenue given the fixed initial inventory of $N$ products. We give a policy that achieves a regret of $\tilde O\left(K \sqrt{K N T}\left(1 + \frac{\sqrt{v_{\max}}}{q_{\min}}\text{OPT}\right) \right)$ under a mild assumption on the model parameters. In particular, our policy achieves a near-optimal $\tilde O(\sqrt{T})$ regret in the large inventory setting. Our policy builds upon the UCB-based approach for MNL-bandit without inventory constraints in [1] and addresses the inventory constraints through an exponentially sized LP for which we present a tractable approximation while keeping the $\tilde O(\sqrt{T})$ regret bound.